Problem 9 - Calculus Volume 1

Question

Use Equation 3.3 to find the slope of the secant line between the values \(x_{1}\) and \(x_{2}\) for each function \(y=f(x)\). $$ f(x)=x^{1 / 3}+1 ; x_{1}=0, x_{2}=8 $$

Step-by-Step Solution

Answer

The slope of the secant line is \( \frac{1}{4} \).

Step 1: Identify the Function and Points

The given function is \( f(x) = x^{1/3} + 1 \). We need to find the slope of the secant line between the points \( x_1 = 0 \) and \( x_2 = 8 \).

Step 2: Calculate the Function Values

Calculate \( f(x_1) \) and \( f(x_2) \):- \( f(x_1) = f(0) = 0^{1/3} + 1 = 1 \)- \( f(x_2) = f(8) = 8^{1/3} + 1 = 2 + 1 = 3 \)

Step 3: Apply the Secant Line Formula

The formula for the slope \( m \) of the secant line between \( x_1 \) and \( x_2 \) is given by:\[m = \frac{f(x_2) - f(x_1)}{x_2 - x_1}\]Substitute the calculated values:\[m = \frac{3 - 1}{8 - 0} = \frac{2}{8} = \frac{1}{4}\]

Step 4: Interpret the Result

The slope of the secant line between \( x = 0 \) and \( x = 8 \) for the function \( f(x) \) is \( \frac{1}{4} \). This represents the average rate of change of the function over the interval.

Key Concepts

Understanding Slope

When you hear the word "slope," think of how steep a line is. A slope tells you how much a line goes up or catches on as you move from one point to another, horizontally. It's like measuring a ramp's tilt. In math, slope is calculated by the change in the vertical direction (Δy), divided by the change in the horizontal direction (Δx). This is possible by using the formula:

• \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

where \( y_2 \) and \( y_1 \) are the values of the function at the points \( x_2 \) and \( x_1 \).
The slope of a secant line between two points on a graph of a function helps indicate how quickly the function's value is changing between these two points.
In the exercise, we found that the slope of the secant line between points \( x=0 \) and \( x=8 \) was \( \frac{1}{4} \). This fraction implies a gentle incline; for every 4 units you move horizontally, the function's value only increases by 1 unit.

Average Rate of Change Demystified

Imagine you are driving a car for a certain amount of time. To find out how fast you went on average, you'd take the total distance driven and divide it by the total time spent driving. The average rate of change is a similar idea but applied to functions.
It tells you, on average, how one quantity is changing with respect to another. In math, when you're looking at a function's graph, this concept is represented by the slope of a secant line.
For the function \( f(x) = x^{1/3} + 1 \) in our exercise, the average rate of change between \( x=0 \) and \( x=8 \) is \( \frac{1}{4} \). This means that, on average, the value of the function increased by 1 unit every time \( x \) increased by 4 units. It's like a snapshot of the function's behavior over a specific interval.

The Role of a Function

Functions are mathematical tools that express relationships between two quantities.
A function takes an input and gives an output, following a specific rule. In the exercise, we had the function \( f(x) = x^{1/3} + 1 \). This function transforms each input \( x \) by raising it to the power of \( \frac{1}{3} \), then adding 1 to the result.
Functions are valuable in different areas of life and science because they allow us to model and predict behaviors and outcomes. Whether predicting population growth or calculating the trajectory of a spacecraft, functions help communicate how one quantity depends on another.
In our specific problem, the function showed us how the y-values changed as we moved from \( x=0 \) to \( x=8 \). Understanding this relationship is fundamental for solving real-world problems.